Course materials for 2020-11-2 AFEC at XTBG.

1 Prerequisites

Did you install picante and FD?

install.packages("picante")
install.packages("FD")

It’s better if you have those packages too.

install.packages("tidyverse")
install.packages("rmarkdown")
install.packages("DT")

Load pacakges.

library(picante)
library(FD)
library(tidyverse)
library(rmarkdown)

2 Data

2.1 Community

samp <- read_csv("./data/samp.csv")

DT::datatable(samp)

samp_mat <- as.matrix(samp[, -1])
rownames(samp_mat) <- samp$Site

samp_mat

##       Illicium_macranthum Manglietia_insignis Michelia_floribunda
## Site1                   1                   0                   0
## Site2                   1                   2                   2
## Site3                   1                   0                   0
## Site4                   1                   1                   0
## Site5                   1                   0                   0
##       Beilschmiedia_robusta Neolitsea_chuii Lindera_thomsonii
## Site1                     0               0                 0
## Site2                     2               0                 0
## Site3                     0               0                 2
## Site4                     0               0                 2
## Site5                     0               1                 1
##       Actinodaphne_forrestii Machilus_yunnanensis
## Site1                      0                    0
## Site2                      0                    0
## Site3                      2                    2
## Site4                      2                    0
## Site5                      0                    0

2.2 Phylogeny

phylo <- read.tree("./data/dummy_tree.newick")
plot(phylo)

2.3 Traits

Abbreviation	Trait	Unit
LMA	Leaf mass per area	g m^-2
LL	Leaf lifespans (longevity)	months
Amass	Maximum photosynthetic rates per unit mass	nnoml g^-1 s^-1
Rmass	Dark resperation rates per unit mass	nnoml g^-1 s^-1
Nmass	Leaf nitrogen per unit mass	%
Pmass	Leaf phosphorus per unit mass	%
WD	Wood density	g cm^-3
SM	Seed dry mass	mg

trait <- read_csv("./data/dummy_trait.csv") 

# trait <- read.csv("./data/dummy_trait.csv") is fine too.

DT::datatable(trait)

2.4 Check the histograms of trait values first

trait_long <- trait %>%
  gather(trait, val, 2:9)

ggplot(trait_long, aes(x = val)) +
  geom_histogram(position = "identity") +
  facet_wrap(~ trait, scale = "free")

Probably we can do log-transformation for all the traits except for WD.

trait2 <- trait %>%
  mutate(logLMA = log(LMA),
         logLL = log(LL),
         logAmass = log(Amass),
         logRmass = log(Rmass),
         logNmass = log(Nmass),
         logPmass = log(Pmass),
         logSM = log(SM)) %>%
  dplyr::select(sp, logLMA, logLL, logAmass, logRmass, logNmass, logPmass, WD, logSM)

DT::datatable(trait2)

trait2 %>%
  gather(trait, val, 2:9) %>%
  ggplot(., aes(x = val)) +
  geom_histogram(position = "identity") +
  facet_wrap(~ trait, scale = "free")

3 Fisrt-order metrics (without phylogeny or traits)

3.1 Species richness, Beta diversity metrics

Skip

3.2 Nonmetric Multidimensional Scaling (NMDS)

res_mds <- metaMDS(samp_mat)

## Run 0 stress 0 
## Run 1 stress 9.744536e-05 
## ... Procrustes: rmse 0.1288544  max resid 0.1986308 
## Run 2 stress 9.489838e-05 
## ... Procrustes: rmse 0.1288521  max resid 0.1986145 
## Run 3 stress 7.044895e-05 
## ... Procrustes: rmse 0.09291819  max resid 0.1402673 
## Run 4 stress 8.23302e-05 
## ... Procrustes: rmse 0.1288677  max resid 0.1986314 
## Run 5 stress 0 
## ... Procrustes: rmse 0.08145113  max resid 0.1100644 
## Run 6 stress 0.2297529 
## Run 7 stress 0 
## ... Procrustes: rmse 0.07308683  max resid 0.1194549 
## Run 8 stress 0.1302441 
## Run 9 stress 0 
## ... Procrustes: rmse 0.04192437  max resid 0.06171832 
## Run 10 stress 0.2297529 
## Run 11 stress 0 
## ... Procrustes: rmse 0.1099842  max resid 0.1878721 
## Run 12 stress 0 
## ... Procrustes: rmse 0.06122222  max resid 0.09081168 
## Run 13 stress 6.968383e-05 
## ... Procrustes: rmse 0.1216506  max resid 0.182077 
## Run 14 stress 0.09680973 
## Run 15 stress 0 
## ... Procrustes: rmse 0.05016184  max resid 0.0750991 
## Run 16 stress 0.1302441 
## Run 17 stress 0.2297529 
## Run 18 stress 0.0968105 
## Run 19 stress 7.031243e-05 
## ... Procrustes: rmse 0.1289714  max resid 0.1987071 
## Run 20 stress 6.946735e-05 
## ... Procrustes: rmse 0.1430172  max resid 0.2603153 
## *** No convergence -- monoMDS stopping criteria:
##     13: stress < smin
##      3: stress ratio > sratmax
##      4: scale factor of the gradient < sfgrmin

plot(res_mds)

We can use the function ordiplot and orditorp to add text to the plot in place of points to make some more sence.

ordiplot(res_mds, type = "n")
orditorp(res_mds,display="species",col="red",air=0.01)
orditorp(res_mds,display="sites",cex=1.25,air=0.01)

4 Phylogenetic metrics

4.1 Branch length based metric

4.1.1 PD

res_pd <- pd(samp_mat, phylo)

res_pd

##             PD SR
## Site1 1.000000  1
## Site2 3.022727  4
## Site3 2.909091  4
## Site4 3.136364  4
## Site5 2.454545  3

You can always see the help.

?pd

4.2 Distance based metric

cophenetic() creates distance matrices based on phylogenetic trees. Let’s see the first 5 species.

cophenetic(phylo)[1:5, 1:5]

##                     Acer_campbellii Melia_toosendan Skimmia_arborescens
## Acer_campbellii           0.0000000      0.18181818          0.18181818
## Melia_toosendan           0.1818182      0.00000000          0.09090909
## Skimmia_arborescens       0.1818182      0.09090909          0.00000000
## Rhus_sylvestris           0.3636364      0.36363636          0.36363636
## Sterculia_nobilis         0.5454545      0.54545455          0.54545455
##                     Rhus_sylvestris Sterculia_nobilis
## Acer_campbellii           0.3636364         0.5454545
## Melia_toosendan           0.3636364         0.5454545
## Skimmia_arborescens       0.3636364         0.5454545
## Rhus_sylvestris           0.0000000         0.5454545
## Sterculia_nobilis         0.5454545         0.0000000

4.2.1 MPD

\(MPD = \frac{1}{n} \Sigma^n_i \Sigma^n_j \delta_{i,j} \; i \neq j\), where \(\delta_{i, j}\) is the pairwised distance between species i and j

res_mpd <- mpd(samp_mat, cophenetic(phylo))

res_mpd

## [1]       NA 1.568182 1.454545 1.606061 1.636364

The above vector shows MPD for each site.

4.2.2 MNTD

\(MNTD = \frac{1}{n} \Sigma^n_i min \delta_{i,j} \; i \neq j\), where \(min \delta_{i, j}\) is the minimum distance between species i and all other species in the community.

res_mntd <- mntd(samp_mat, cophenetic(phylo))

res_mntd

## [1]       NA 1.181818 1.181818 1.295455 1.272727

5 Functional metrics

5.1 Community weighted means (CWM)

\[ CWM_i = \frac{\sum_{j=1}^n a_{ij} \times t_{j}}{\sum_{j=1}^n a_{ij}} \]

tmp <- trait2 %>%
  filter(sp %in% colnames(samp_mat)) 

tmp

## # A tibble: 8 x 9
##   sp              logLMA logLL logAmass logRmass logNmass logPmass    WD   logSM
##   <chr>            <dbl> <dbl>    <dbl>    <dbl>    <dbl>    <dbl> <dbl>   <dbl>
## 1 Actinodaphne_f…   4.24  2.53     5.01    2.17     0.412   -1.83   0.48  0.300 
## 2 Beilschmiedia_…   3.61  3.09     5.72    3.53     1.75    -1.35   0.47  0.770 
## 3 Illicium_macra…   5.66  4.75     3.27    0.793   -0.288   -3.51   0.4  -0.0305
## 4 Lindera_thomso…   4.47  3.70     5.49    3.02     0.626   -3.00   0.53 -0.734 
## 5 Machilus_yunna…   4.26  3.36     4.65    2.69     0.239   -0.821  0.59  0.0770
## 6 Manglietia_ins…   6.22  5.24     3.10    0.255   -0.431   -3.91   0.45 -0.0513
## 7 Michelia_flori…   4.93  3.99     3.65    2.00     0.457   -3.91   0.54  0.621 
## 8 Neolitsea_chuii   4.65  4.18     5.20    2.30     0.489   -2.12   0.43 -1.71

(ab <- apply(samp_mat, 1, sum))

## Site1 Site2 Site3 Site4 Site5 
##     1     7     7     6     3

# inner product
(CWS <- samp_mat %*% as.matrix(tmp[,-1]))

##          logLMA     logLL  logAmass  logRmass  logNmass   logPmass   WD
## Site1  4.236712  2.527327  5.006359  2.173615 0.4121097  -1.832581 0.48
## Site2 31.729450 25.585161 33.973907 16.848875 4.5974297 -17.531309 3.28
## Site3 35.828159 29.342331 28.910240 11.266140 1.4425535 -21.721201 3.32
## Site4 30.140733 24.069613 24.233478 10.201674 2.2197972 -18.827747 2.93
## Site5 14.713415 11.128090 12.759265  5.116104 0.2203436  -6.565585 1.52
##            logSM
## Site1  0.3001046
## Site2  0.3114643
## Site3 -1.9909259
## Site4  2.2087792
## Site5  0.3257723

(CWM <- CWS / ab)

##         logLMA    logLL logAmass logRmass   logNmass  logPmass        WD
## Site1 4.236712 2.527327 5.006359 2.173615 0.41210965 -1.832581 0.4800000
## Site2 4.532779 3.655023 4.853415 2.406982 0.65677568 -2.504473 0.4685714
## Site3 5.118308 4.191762 4.130034 1.609449 0.20607908 -3.103029 0.4742857
## Site4 5.023456 4.011602 4.038913 1.700279 0.36996620 -3.137958 0.4883333
## Site5 4.904472 3.709363 4.253088 1.705368 0.07344788 -2.188528 0.5066667
##            logSM
## Site1  0.3001046
## Site2  0.0444949
## Site3 -0.2844180
## Site4  0.3681299
## Site5  0.1085908

5.2 Distance based metrics

5.2.1 Prepare a trait distance matrix

We have a data.fame of traits. First we need to prepare a trait matrix, then a distance matrix based on trait values.

trait_mat0 <- as.matrix(trait2[, -1])
rownames(trait_mat0) <- trait2$sp

Let’s see a subset of the trait matrix

trait_mat0[1:5, 1:5]

##                          logLMA    logLL logAmass logRmass  logNmass
## Acer_campbellii        3.684118 1.957274 6.892692 4.002047 1.8809906
## Actinodaphne_forrestii 4.236712 2.527327 5.006359 2.173615 0.4121097
## Alnus_nepalensis       4.743366 4.010419 4.341335 2.022871 0.5007753
## Anneslea_fragrans      4.190715 3.293241 5.162211 3.703522 1.4632554
## Beilschmiedia_robusta  3.614964 3.085573 5.722441 3.526655 1.7544037

Then, we will make trait distance matrix based on the Euclidean distance. There are other distance measures, for example Gower’s Distance, but we focus on the Euclidean distance today.

Before calulating distance, we need to make sure unit change in ditances have same for different traits. We will scale trait values so that then have mean = 0 and SD = 1. (e.g., \((X_i - \mu) / \sigma\))

trait_mat <- scale(trait_mat0)

par(mfrow = c(2, 2))
hist(trait_mat0[, "logLMA"])
hist(trait_mat[, "logLMA"])
hist(trait_mat0[, "WD"])
hist(trait_mat[, "WD"])

par(mfrow = c(1, 1))

Now we can make a trait distance matirx.

trait_dm <- as.matrix(dist(trait_mat))

Let’s see the first 5 species.

trait_dm[1:5, 1:5]

##                        Acer_campbellii Actinodaphne_forrestii Alnus_nepalensis
## Acer_campbellii               0.000000               3.799360         5.216902
## Actinodaphne_forrestii        3.799360               0.000000         2.415031
## Alnus_nepalensis              5.216902               2.415031         0.000000
## Anneslea_fragrans             3.175911               2.335392         3.225141
## Beilschmiedia_robusta         2.545269               2.565063         3.638183
##                        Anneslea_fragrans Beilschmiedia_robusta
## Acer_campbellii                 3.175911              2.545269
## Actinodaphne_forrestii          2.335392              2.565063
## Alnus_nepalensis                3.225141              3.638183
## Anneslea_fragrans               0.000000              1.579930
## Beilschmiedia_robusta           1.579930              0.000000

5.2.2 MPD

mpd(samp_mat, trait_dm)

## [1]       NA 4.288349 3.530805 3.961248 3.438008

ses.mpd(samp_mat, trait_dm)

##       ntaxa  mpd.obs mpd.rand.mean mpd.rand.sd mpd.obs.rank  mpd.obs.z
## Site1     1       NA           NaN          NA           NA         NA
## Site2     4 4.288349      3.718044   0.7981511          761  0.7145337
## Site3     4 3.530805      3.737433   0.7862713          428 -0.2627945
## Site4     4 3.961248      3.711583   0.7912637          657  0.3155273
## Site5     3 3.438008      3.751037   0.9857789          420 -0.3175449
##       mpd.obs.p runs
## Site1        NA  999
## Site2     0.761  999
## Site3     0.428  999
## Site4     0.657  999
## Site5     0.420  999

5.2.3 MNTD

mntd(samp_mat, trait_dm)

## [1]       NA 2.504352 2.697074 1.873825 2.613585

5.3 Branch length based metric

5.3.1 FD

We will make a functional dendrogram using clustring methods. We use UPGMA in this example.

t_clust <- hclust(dist(trait_mat), method = "average")

plot(t_clust)

5.3.2 More functional diversity metrics

res_fd <- dbFD(trait_mat[colnames(samp_mat), ], samp_mat)

## FEVe: Could not be calculated for communities with <3 functionally singular species. 
## FDis: Equals 0 in communities with only one functionally singular species. 
## FRic: To respect s > t, FRic could not be calculated for communities with <3 functionally singular species. 
## FRic: Dimensionality reduction was required. The last 5 PCoA axes (out of 7 in total) were removed. 
## FRic: Quality of the reduced-space representation = 0.811349 
## FDiv: Could not be calculated for communities with <3 functionally singular species.

res_fd

## $nbsp
## Site1 Site2 Site3 Site4 Site5 
##     1     4     4     4     3 
## 
## $sing.sp
## Site1 Site2 Site3 Site4 Site5 
##     1     4     4     4     3 
## 
## $FRic
##    Site1    Site2    Site3    Site4    Site5 
##       NA 5.453089 2.917904 3.000656 3.553247 
## 
## $qual.FRic
## [1] 0.811349
## 
## $FEve
##     Site1     Site2     Site3     Site4     Site5 
##        NA 0.7595456 0.6769400 0.7085376 0.7584941 
## 
## $FDiv
##     Site1     Site2     Site3     Site4     Site5 
##        NA 0.7301943 0.7617251 0.9166699 0.8261683 
## 
## $FDis
##    Site1    Site2    Site3    Site4    Site5 
## 0.000000 2.710994 1.842262 2.311159 2.042416 
## 
## $RaoQ
##    Site1    Site2    Site3    Site4    Site5 
## 0.000000 8.376023 4.005094 5.664467 4.379844 
## 
## $CWM
##           logLMA       logLL    logAmass   logRmass    logNmass   logPmass
## Site1  1.4467783  1.17548950 -1.38976382 -1.9975087 -0.88119735 -1.2775781
## Site2  0.5666449  0.55085046 -0.56218769 -0.8908026 -0.09004842 -0.8660119
## Site3 -0.1410729 -0.33319385  0.27087040 -0.2062427 -0.24084641  0.2088166
## Site4  0.3670613  0.03104745  0.01551229 -0.7298853 -0.34295985 -0.5718506
## Site5  0.4305791  0.56352114  0.11718014 -0.5812855 -0.29128834 -0.6020974
##               WD      logSM
## Site1 -1.0150179 -0.2191496
## Site2 -0.2744691  0.1665816
## Site3  0.1242879 -0.2907346
## Site4 -0.2341187 -0.3397288
## Site5 -0.4833418 -0.9701997

6 Computing Environment

devtools::session_info()

## ─ Session info ───────────────────────────────────────────────────────────────
##  setting  value                       
##  version  R version 4.0.2 (2020-06-22)
##  os       Ubuntu 20.04 LTS            
##  system   x86_64, linux-gnu           
##  ui       X11                         
##  language (EN)                        
##  collate  en_US.UTF-8                 
##  ctype    en_US.UTF-8                 
##  tz       Etc/UTC                     
##  date     2020-11-01                  
## 
## ─ Packages ───────────────────────────────────────────────────────────────────
##  package     * version  date       lib source        
##  abind         1.4-5    2016-07-21 [1] RSPM (R 4.0.0)
##  ade4        * 1.7-15   2020-02-13 [1] RSPM (R 4.0.0)
##  ape         * 5.4-1    2020-08-13 [1] RSPM (R 4.0.2)
##  assertthat    0.2.1    2019-03-21 [1] RSPM (R 4.0.0)
##  backports     1.1.8    2020-06-17 [1] RSPM (R 4.0.1)
##  blob          1.2.1    2020-01-20 [1] RSPM (R 4.0.0)
##  broom         0.7.0    2020-07-09 [1] RSPM (R 4.0.2)
##  callr         3.4.3    2020-03-28 [1] RSPM (R 4.0.0)
##  cellranger    1.1.0    2016-07-27 [1] RSPM (R 4.0.0)
##  cli           2.0.2    2020-02-28 [1] RSPM (R 4.0.0)
##  cluster       2.1.0    2019-06-19 [2] CRAN (R 4.0.2)
##  colorspace    1.4-1    2019-03-18 [1] RSPM (R 4.0.0)
##  crayon        1.3.4    2017-09-16 [1] RSPM (R 4.0.0)
##  crosstalk     1.1.0.1  2020-03-13 [1] RSPM (R 4.0.0)
##  DBI           1.1.0    2019-12-15 [1] RSPM (R 4.0.0)
##  dbplyr        1.4.4    2020-05-27 [1] RSPM (R 4.0.0)
##  desc          1.2.0    2018-05-01 [1] RSPM (R 4.0.0)
##  devtools      2.3.0    2020-04-10 [1] RSPM (R 4.0.0)
##  digest        0.6.25   2020-02-23 [1] RSPM (R 4.0.0)
##  dplyr       * 1.0.0    2020-05-29 [1] RSPM (R 4.0.0)
##  DT            0.14     2020-06-24 [1] RSPM (R 4.0.2)
##  ellipsis      0.3.1    2020-05-15 [1] RSPM (R 4.0.0)
##  evaluate      0.14     2019-05-28 [1] RSPM (R 4.0.0)
##  fansi         0.4.1    2020-01-08 [1] RSPM (R 4.0.0)
##  farver        2.0.3    2020-01-16 [1] RSPM (R 4.0.0)
##  FD          * 1.0-12   2014-08-19 [1] RSPM (R 4.0.0)
##  forcats     * 0.5.0    2020-03-01 [1] RSPM (R 4.0.0)
##  fs            1.4.2    2020-06-30 [1] RSPM (R 4.0.2)
##  generics      0.0.2    2018-11-29 [1] RSPM (R 4.0.0)
##  geometry    * 0.4.5    2019-12-04 [1] RSPM (R 4.0.2)
##  ggplot2     * 3.3.2    2020-06-19 [1] RSPM (R 4.0.1)
##  glue          1.4.1    2020-05-13 [1] RSPM (R 4.0.0)
##  gtable        0.3.0    2019-03-25 [1] RSPM (R 4.0.0)
##  haven         2.3.1    2020-06-01 [1] RSPM (R 4.0.2)
##  hms           0.5.3    2020-01-08 [1] RSPM (R 4.0.0)
##  htmltools     0.5.0    2020-06-16 [1] RSPM (R 4.0.1)
##  htmlwidgets   1.5.1    2019-10-08 [1] RSPM (R 4.0.0)
##  httr          1.4.1    2019-08-05 [1] RSPM (R 4.0.0)
##  jsonlite      1.7.0    2020-06-25 [1] RSPM (R 4.0.2)
##  knitr         1.29     2020-06-23 [1] RSPM (R 4.0.2)
##  labeling      0.3      2014-08-23 [1] RSPM (R 4.0.0)
##  lattice     * 0.20-41  2020-04-02 [2] CRAN (R 4.0.2)
##  lifecycle     0.2.0    2020-03-06 [1] RSPM (R 4.0.0)
##  lubridate     1.7.9    2020-06-08 [1] RSPM (R 4.0.2)
##  magic         1.5-9    2018-09-17 [1] RSPM (R 4.0.0)
##  magrittr      1.5      2014-11-22 [1] RSPM (R 4.0.0)
##  MASS          7.3-51.6 2020-04-26 [2] CRAN (R 4.0.2)
##  Matrix        1.2-18   2019-11-27 [2] CRAN (R 4.0.2)
##  memoise       1.1.0    2017-04-21 [1] RSPM (R 4.0.0)
##  mgcv          1.8-31   2019-11-09 [2] CRAN (R 4.0.2)
##  modelr        0.1.8    2020-05-19 [1] RSPM (R 4.0.0)
##  munsell       0.5.0    2018-06-12 [1] RSPM (R 4.0.0)
##  nlme        * 3.1-148  2020-05-24 [2] CRAN (R 4.0.2)
##  permute     * 0.9-5    2019-03-12 [1] RSPM (R 4.0.0)
##  picante     * 1.8.2    2020-06-10 [1] RSPM (R 4.0.0)
##  pillar        1.4.6    2020-07-10 [1] RSPM (R 4.0.2)
##  pkgbuild      1.1.0    2020-07-13 [1] RSPM (R 4.0.2)
##  pkgconfig     2.0.3    2019-09-22 [1] RSPM (R 4.0.0)
##  pkgload       1.1.0    2020-05-29 [1] RSPM (R 4.0.0)
##  prettyunits   1.1.1    2020-01-24 [1] RSPM (R 4.0.0)
##  processx      3.4.3    2020-07-05 [1] RSPM (R 4.0.2)
##  ps            1.3.3    2020-05-08 [1] RSPM (R 4.0.0)
##  purrr       * 0.3.4    2020-04-17 [1] RSPM (R 4.0.0)
##  R6            2.4.1    2019-11-12 [1] RSPM (R 4.0.0)
##  Rcpp          1.0.5    2020-07-06 [1] RSPM (R 4.0.2)
##  readr       * 1.3.1    2018-12-21 [1] RSPM (R 4.0.2)
##  readxl        1.3.1    2019-03-13 [1] RSPM (R 4.0.2)
##  remotes       2.1.1    2020-02-15 [1] RSPM (R 4.0.0)
##  reprex        0.3.0    2019-05-16 [1] RSPM (R 4.0.0)
##  rlang         0.4.7    2020-07-09 [1] RSPM (R 4.0.2)
##  rmarkdown   * 2.3      2020-06-18 [1] RSPM (R 4.0.1)
##  rprojroot     1.3-2    2018-01-03 [1] RSPM (R 4.0.0)
##  rstudioapi    0.11     2020-02-07 [1] RSPM (R 4.0.0)
##  rvest         0.3.5    2019-11-08 [1] RSPM (R 4.0.0)
##  scales        1.1.1    2020-05-11 [1] RSPM (R 4.0.0)
##  sessioninfo   1.1.1    2018-11-05 [1] RSPM (R 4.0.0)
##  stringi       1.4.6    2020-02-17 [1] RSPM (R 4.0.0)
##  stringr     * 1.4.0    2019-02-10 [1] RSPM (R 4.0.0)
##  testthat      2.3.2    2020-03-02 [1] RSPM (R 4.0.0)
##  tibble      * 3.0.3    2020-07-10 [1] RSPM (R 4.0.2)
##  tidyr       * 1.1.0    2020-05-20 [1] RSPM (R 4.0.2)
##  tidyselect    1.1.0    2020-05-11 [1] RSPM (R 4.0.0)
##  tidyverse   * 1.3.0    2019-11-21 [1] RSPM (R 4.0.0)
##  usethis       1.6.1    2020-04-29 [1] RSPM (R 4.0.0)
##  utf8          1.1.4    2018-05-24 [1] RSPM (R 4.0.0)
##  vctrs         0.3.2    2020-07-15 [1] RSPM (R 4.0.2)
##  vegan       * 2.5-6    2019-09-01 [1] RSPM (R 4.0.0)
##  withr         2.2.0    2020-04-20 [1] RSPM (R 4.0.0)
##  xfun          0.15     2020-06-21 [1] RSPM (R 4.0.1)
##  xml2          1.3.2    2020-04-23 [1] RSPM (R 4.0.0)
##  yaml          2.2.1    2020-02-01 [1] RSPM (R 4.0.0)
## 
## [1] /usr/local/lib/R/site-library
## [2] /usr/local/lib/R/library

Diversity calculation

Masatoshi Katabuchi

2020/11/2 (updated: 2020-11-01)